## Periodic Reporting for period 2 - HHQM (Hydrodynamics, holography and strongly-coupled quantum matter)

Reporting period: 2020-03-01 to 2021-08-31

Describing states of matter starting from their microscopic constituents is often a very hard task. If the interactions between the microscopic constituents are strong, as is the case in Quantum ChromoDynamics at low energies or in strongly-correlated quantum materials, perturbation theory fails for lack of a small expansion parameter. If they are weak, exact solutions are out of reach in quantum many-body problems: while conceptually straightforward, diagonalizing Hamiltonians for a large number of particles is in practice impossible. Effective theories and dualities are two of the tools at our disposal which circumvent these problems, each in their own specific way.

Effective theories use symmetries as an organizing principle to describe the dynamics of a given system at times and distances very long compared to some cut-off scale. This cut-off is often set by temperature and corresponds to the scale at which local equilibrium is established. Effective equations can then be written down as an expansion in spatial and temporal derivatives, controlled by inverse powers of the cut-off. The transport coefficients in front of each of these terms are not determined by the effective theory, and instead need to be computed on a case-by-case basis, which is also often a hard task.

One of the best known examples of such an effective theory is hydrodynamics, which describes the collective dynamics of water flows in terms of only a few variables (temperature, fluid velocity, chemical potential) together with a finite number of transport coefficients, to be contrasted with the vast number of individual water molecules making up the flow. The theory relies on the conservation of energy, momentum and charge, following from invariance under translations, rotations, Galilean boosts and conservation of particle number.

Fluid hydrodynamics retains a very large amount of symmetries. Many interesting phases of matter only exhibit a subset of these symmetries. A significant amount of my recent research, as well as some of my future research plans, are devoted to constructing effective theories in systems with reduced symmetries, and applying them to various phases of matter, such as Graphene, high critical temperature superconductors and so-called bad metals. Bad metals are so badly conducting that their large resistivities are incompatible with weakly-coupled electrons. Moreover, they are also often strange, with a resistivity linear in temperature, rather than quadratic as would follow from weakly-coupled electrons.

Dualities are another extremely powerful tool. A duality states that Theory 1 is equivalent (at the quantum level) to Theory 2. This is especially interesting when the duality holds in a regime where Theory 1 is difficult to solve, but Theory 2 is not. This is the case for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, which has revolutionized String theory and High Energy Quantum Field Theory in the last two decades.

Anti-de Sitter spacetimes are maximally symmetric solutions to Einstein's equations with a negative cosmological constant. Conformal Field Theories have a larger symmetry group than the usual Poincaré: they also contain rigid scale transformations as well as special conformal transformations, which are a certain kind of angle-preserving. It turns out that anti de Sitter spacetimes have a `boundary', and that the isometries of the metric are precisely those of a Conformal Field Theory in one less dimension.

A precise version of the AdS/CFT duality (also known as Holography) can be found in String Theory. For our purposes, it suffices to say that weakly-coupled gravity theories on an asymptotically anti de Sitter background are dual to certain strongly-coupled Quantum Field Theories which are Conformal Field Theories at high energies.

At first sight, this would appear to place severe limitations on the usefulness of the AdS/CFT duality. After all, most physical systems are not Conformal Field Theories in the ultra-violet. However, there are two useful ways to deform the Conformal Field Theory.

Firstly, by turning on a symmetry-breaking deformation, the system will be described in the infra-red by an effective field theory with a reduced amount of symmetry, eg: fluids with weak momentum relaxation, superfluids, charge density waves, etc. This is especially interesting when the symmetry is explicitly broken. Indeed, while there is a clear strategy for writing down effective field theories in the hydrodynamic limit based on an expansion in derivatives around equilibrium and taking into account the symmetries of the system, symmetry-breaking terms fall outside this strategy and are typically introduced `by hand' in the effective field theory. Gauge-gravity duality provides a microscopically-complete framework in which the hydrodynamic limit can be taken consistently and the corresponding effective field theory derived.

Secondly, one can deform the holographic ultra-violet Conformal Field Theory to trigger a (renormalization group) flow towards a zero temperature, infra-red fixed point – a new theory. Strongly-correlated Condensed Matter systems often sport a quantum critical point, which separates an ordered from a disordered phase at zero temperature and mediates a quantum phase transition as some external parameter is varied (such as the magnetic field, the pressure, the chemical composition...). At the quantum critical point, there is an emergent scale invariance and an associated emergent infra-red effective field theory. Gauge-gravity duality allows a straightforward modeling of this quantum critical phase.

Blending together the effective field theory and holographic approaches to study strongly-coupled quantum phases of matter is the heart of this ERC project, carried out by the research group of Dr. Blaise Goutéraux at the Center for Theoretical Physics of Ecole Polytechnique, Palaiseau, France. More specifically, we will focus on the interplay between momentum relaxation by translation-breaking effects and quantum criticality; the rôle played by fundamental bounds on transport ; phase-relaxing effects in system where a symmetry is spontaneously broken, including superfluids and charge density waves.

Effective theories use symmetries as an organizing principle to describe the dynamics of a given system at times and distances very long compared to some cut-off scale. This cut-off is often set by temperature and corresponds to the scale at which local equilibrium is established. Effective equations can then be written down as an expansion in spatial and temporal derivatives, controlled by inverse powers of the cut-off. The transport coefficients in front of each of these terms are not determined by the effective theory, and instead need to be computed on a case-by-case basis, which is also often a hard task.

One of the best known examples of such an effective theory is hydrodynamics, which describes the collective dynamics of water flows in terms of only a few variables (temperature, fluid velocity, chemical potential) together with a finite number of transport coefficients, to be contrasted with the vast number of individual water molecules making up the flow. The theory relies on the conservation of energy, momentum and charge, following from invariance under translations, rotations, Galilean boosts and conservation of particle number.

Fluid hydrodynamics retains a very large amount of symmetries. Many interesting phases of matter only exhibit a subset of these symmetries. A significant amount of my recent research, as well as some of my future research plans, are devoted to constructing effective theories in systems with reduced symmetries, and applying them to various phases of matter, such as Graphene, high critical temperature superconductors and so-called bad metals. Bad metals are so badly conducting that their large resistivities are incompatible with weakly-coupled electrons. Moreover, they are also often strange, with a resistivity linear in temperature, rather than quadratic as would follow from weakly-coupled electrons.

Dualities are another extremely powerful tool. A duality states that Theory 1 is equivalent (at the quantum level) to Theory 2. This is especially interesting when the duality holds in a regime where Theory 1 is difficult to solve, but Theory 2 is not. This is the case for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, which has revolutionized String theory and High Energy Quantum Field Theory in the last two decades.

Anti-de Sitter spacetimes are maximally symmetric solutions to Einstein's equations with a negative cosmological constant. Conformal Field Theories have a larger symmetry group than the usual Poincaré: they also contain rigid scale transformations as well as special conformal transformations, which are a certain kind of angle-preserving. It turns out that anti de Sitter spacetimes have a `boundary', and that the isometries of the metric are precisely those of a Conformal Field Theory in one less dimension.

A precise version of the AdS/CFT duality (also known as Holography) can be found in String Theory. For our purposes, it suffices to say that weakly-coupled gravity theories on an asymptotically anti de Sitter background are dual to certain strongly-coupled Quantum Field Theories which are Conformal Field Theories at high energies.

At first sight, this would appear to place severe limitations on the usefulness of the AdS/CFT duality. After all, most physical systems are not Conformal Field Theories in the ultra-violet. However, there are two useful ways to deform the Conformal Field Theory.

Firstly, by turning on a symmetry-breaking deformation, the system will be described in the infra-red by an effective field theory with a reduced amount of symmetry, eg: fluids with weak momentum relaxation, superfluids, charge density waves, etc. This is especially interesting when the symmetry is explicitly broken. Indeed, while there is a clear strategy for writing down effective field theories in the hydrodynamic limit based on an expansion in derivatives around equilibrium and taking into account the symmetries of the system, symmetry-breaking terms fall outside this strategy and are typically introduced `by hand' in the effective field theory. Gauge-gravity duality provides a microscopically-complete framework in which the hydrodynamic limit can be taken consistently and the corresponding effective field theory derived.

Secondly, one can deform the holographic ultra-violet Conformal Field Theory to trigger a (renormalization group) flow towards a zero temperature, infra-red fixed point – a new theory. Strongly-correlated Condensed Matter systems often sport a quantum critical point, which separates an ordered from a disordered phase at zero temperature and mediates a quantum phase transition as some external parameter is varied (such as the magnetic field, the pressure, the chemical composition...). At the quantum critical point, there is an emergent scale invariance and an associated emergent infra-red effective field theory. Gauge-gravity duality allows a straightforward modeling of this quantum critical phase.

Blending together the effective field theory and holographic approaches to study strongly-coupled quantum phases of matter is the heart of this ERC project, carried out by the research group of Dr. Blaise Goutéraux at the Center for Theoretical Physics of Ecole Polytechnique, Palaiseau, France. More specifically, we will focus on the interplay between momentum relaxation by translation-breaking effects and quantum criticality; the rôle played by fundamental bounds on transport ; phase-relaxing effects in system where a symmetry is spontaneously broken, including superfluids and charge density waves.

*Phases breaking translations spontaneously:

Charge density waves and other phases where translations are spontaneously broken abound in the phase diagram of strongly-correlated materials. The hydrodynamic description of these states is textbook material. Spontaneous symmetry breaking generates a new degree of freedom, the so-called Nambu-Goldstone boson, which couples to the other hydrodynamic variables. Here it represents the freedom to slide the density wave around in the material without energy cost. Disorder is known to `pin’ the Goldstone mode by giving it a small mass. We set out to verify the effective field theory of pinned charge density waves just described using gauge-gravity duality.

Our key finding in this work is that pinning does not simply generate a mass for the Goldstone, but also a decay rate. Moreover, we also derived that in this relaxed holographic system, the ratio of the phonon decay rate and the phonon mass squared is simply given by the phonon diffusivity in the unrelaxed theory. In other words, this is a universal effect independent to leading order on the precise details of disorder.

We have also investigated the hydrodynamic theory of charge density waves in a magnetic field and demonstrated it gives an excellent account of available experimental data on Gallium Arsenide heterostructures, where such states are formed at high magnetic fields.

*Superfluids:

In some phases of matter like liquid Helium, the conservation of particle number is spontaneously broken at low temperature, forming a superfluid. The energy cost of fluctuations of the Goldstone mode is characterized by a stiffness constant (as in a spring), called the superfluid density. In the Bardeen-Cooper-Schrieffer theory of conventional superconductors, this quantity is known to be equal to the total density of charge carriers at zero temperature. The superfluid density has recently been measured in the overdoped part of the phase diagram of high Tc superconductors. In this region, these materials are believed to be weakly-coupled and well described by a combination of BCS theory in the superconducting phase, and Fermi liquid theory for the normal phase at very large doping. Yet, these measurements called this picture in question by reporting an anomalously low superfluid density, incompatible with BCS theory predictions. This was interpreted as the result of a non-vanishing residual density of uncondensed electrons at zero temperature.

In our work, we studied strongly-coupled superfluids in the vicinity of a quantum critical point using gauge-gravity duality. We discovered that the scaling properties of the quantum critical point play a key rôle in determining whether the normal density vanishes or not at zero temperature, and constructed many examples where it does not. Our results may help shed some light on the corresponding measurements in overdoped cuprates.

* Breakdown of hydrodynamics and fundamental bounds on transport coefficients :

The breakdown of hydrodynamics at scales of order the local equilibration scale is expected to depend on the microscopic details of the system in the ultra-violet. On the other hand, the dynamics near a quantum critical point which governs a zero temperature quantum phase transition is expected to be universal even at nonzero temperature, in the quantum critical nonzero temperature wedge.

We have examined the breakdown of hydrodynamics near a quantum critical point using gauge-gravity duality. We find that it is controlled by the quantum critical degrees of freedom of the universal infra-red theory. Specifically, the equilibration time τ=h/(4πΔkBT), where T is temperature, kB the Boltzmann constant, h the Planck constant and Δ the scaling dimension of the least irrelevant deformation away from the quantum critical point. In this sense, the breakdown of hydrodynamics is universal since it is no longer dependent on the microscopic details of the system.

Bounds on transport coefficients are of great import, as these quantities are not always easy to calculate microscopically. Such bounds may originate from causality, from consistency with Quantum Mechanics, etc. The Kovtun-Son-Starinets bound on the ratio of the shear viscosity over entropy density has played a key rôle in our qualitative understanding of strongly-coupled quantum phases, such as the Quark-Gluon-Plasma. It can be reformulated as a Quantum bound on the diffusivity of transverse momentum. Similar bounds have been conjectured for other transport coefficients, such as the thermal diffusivity. These bounds usually involved a timescale and a distinguished velocity. In the holographic system we studied, we gave a precise identification of the thermal diffusivity and the diffusivity of transverse momentum in terms of the local equilibration time and length scales, which saturate the bounds mentioned above.

Our work paves the way for a deeper understanding of low temperature transport near a quantum critical point, which is relevant both for High Energy (Quark-Gluon-Plasma) and Condensed Matter Physics (ultra-cold atomic systems, strongly-correlated electronic phases of matter).

Charge density waves and other phases where translations are spontaneously broken abound in the phase diagram of strongly-correlated materials. The hydrodynamic description of these states is textbook material. Spontaneous symmetry breaking generates a new degree of freedom, the so-called Nambu-Goldstone boson, which couples to the other hydrodynamic variables. Here it represents the freedom to slide the density wave around in the material without energy cost. Disorder is known to `pin’ the Goldstone mode by giving it a small mass. We set out to verify the effective field theory of pinned charge density waves just described using gauge-gravity duality.

Our key finding in this work is that pinning does not simply generate a mass for the Goldstone, but also a decay rate. Moreover, we also derived that in this relaxed holographic system, the ratio of the phonon decay rate and the phonon mass squared is simply given by the phonon diffusivity in the unrelaxed theory. In other words, this is a universal effect independent to leading order on the precise details of disorder.

We have also investigated the hydrodynamic theory of charge density waves in a magnetic field and demonstrated it gives an excellent account of available experimental data on Gallium Arsenide heterostructures, where such states are formed at high magnetic fields.

*Superfluids:

In some phases of matter like liquid Helium, the conservation of particle number is spontaneously broken at low temperature, forming a superfluid. The energy cost of fluctuations of the Goldstone mode is characterized by a stiffness constant (as in a spring), called the superfluid density. In the Bardeen-Cooper-Schrieffer theory of conventional superconductors, this quantity is known to be equal to the total density of charge carriers at zero temperature. The superfluid density has recently been measured in the overdoped part of the phase diagram of high Tc superconductors. In this region, these materials are believed to be weakly-coupled and well described by a combination of BCS theory in the superconducting phase, and Fermi liquid theory for the normal phase at very large doping. Yet, these measurements called this picture in question by reporting an anomalously low superfluid density, incompatible with BCS theory predictions. This was interpreted as the result of a non-vanishing residual density of uncondensed electrons at zero temperature.

In our work, we studied strongly-coupled superfluids in the vicinity of a quantum critical point using gauge-gravity duality. We discovered that the scaling properties of the quantum critical point play a key rôle in determining whether the normal density vanishes or not at zero temperature, and constructed many examples where it does not. Our results may help shed some light on the corresponding measurements in overdoped cuprates.

* Breakdown of hydrodynamics and fundamental bounds on transport coefficients :

The breakdown of hydrodynamics at scales of order the local equilibration scale is expected to depend on the microscopic details of the system in the ultra-violet. On the other hand, the dynamics near a quantum critical point which governs a zero temperature quantum phase transition is expected to be universal even at nonzero temperature, in the quantum critical nonzero temperature wedge.

We have examined the breakdown of hydrodynamics near a quantum critical point using gauge-gravity duality. We find that it is controlled by the quantum critical degrees of freedom of the universal infra-red theory. Specifically, the equilibration time τ=h/(4πΔkBT), where T is temperature, kB the Boltzmann constant, h the Planck constant and Δ the scaling dimension of the least irrelevant deformation away from the quantum critical point. In this sense, the breakdown of hydrodynamics is universal since it is no longer dependent on the microscopic details of the system.

Bounds on transport coefficients are of great import, as these quantities are not always easy to calculate microscopically. Such bounds may originate from causality, from consistency with Quantum Mechanics, etc. The Kovtun-Son-Starinets bound on the ratio of the shear viscosity over entropy density has played a key rôle in our qualitative understanding of strongly-coupled quantum phases, such as the Quark-Gluon-Plasma. It can be reformulated as a Quantum bound on the diffusivity of transverse momentum. Similar bounds have been conjectured for other transport coefficients, such as the thermal diffusivity. These bounds usually involved a timescale and a distinguished velocity. In the holographic system we studied, we gave a precise identification of the thermal diffusivity and the diffusivity of transverse momentum in terms of the local equilibration time and length scales, which saturate the bounds mentioned above.

Our work paves the way for a deeper understanding of low temperature transport near a quantum critical point, which is relevant both for High Energy (Quark-Gluon-Plasma) and Condensed Matter Physics (ultra-cold atomic systems, strongly-correlated electronic phases of matter).

*Effective field theories of quantum matter :

There is a growing need to put on firmer footing effective theories of quantum matter where symmetries are explicitly broken and the usual strategy of setting up an expansion in spatial and temporal gradients according to the symmetries of the system no longer operates. The recent development of Schwinger-Keldysh effective theories for hydrodynamics offers the exciting possibility to make progress on this problem. One application of such an endeavour would be a proof of the relation between the damping and the mass of the Goldstone of broken translations discovered earlier over the course of this ERC project.

More generally, we plan on continuing to investigate how the blending of the effective theory and gauge-gravity duality approaches can inform us on experimental results in strongly-correlated Condensed Matter systems.

*Breakdown of hydrodynamics and fundamental bounds on transport coefficients :

We plan on continuing our investigations of quantum critical transport and fundamental bounds, in particular extending our recent analysis to other types of quantum critical points points, such as those with anisotropic scaling between time and space, relevant for cold atomic systems (Fermi gas near unitarity). Diffusive transport has been measured in several experiments on ultracold atoms or high Tc superconductors and may be able to report in the near future on the breakdown of hydrodynamics.

There is a growing need to put on firmer footing effective theories of quantum matter where symmetries are explicitly broken and the usual strategy of setting up an expansion in spatial and temporal gradients according to the symmetries of the system no longer operates. The recent development of Schwinger-Keldysh effective theories for hydrodynamics offers the exciting possibility to make progress on this problem. One application of such an endeavour would be a proof of the relation between the damping and the mass of the Goldstone of broken translations discovered earlier over the course of this ERC project.

More generally, we plan on continuing to investigate how the blending of the effective theory and gauge-gravity duality approaches can inform us on experimental results in strongly-correlated Condensed Matter systems.

*Breakdown of hydrodynamics and fundamental bounds on transport coefficients :

We plan on continuing our investigations of quantum critical transport and fundamental bounds, in particular extending our recent analysis to other types of quantum critical points points, such as those with anisotropic scaling between time and space, relevant for cold atomic systems (Fermi gas near unitarity). Diffusive transport has been measured in several experiments on ultracold atoms or high Tc superconductors and may be able to report in the near future on the breakdown of hydrodynamics.